Hybrid Finite Element and Finite Volume Methods for Two Immiscible Fluid Flows

Abstract

We have successfully extended our implicit hybrid finite element/volume solver to flows involving two immiscible fluids. The solver is based on the segregated pressure correction or projection method on staggered unstructured hybrid meshes. An intermediate velocity field is first obtained by solving the momentum equations with the matrix-free implicit cell-centered finite volume method. The pressure Poisson equation is solved by the node-based Galerkin finite element method for an auxiliary variable. The auxiliary variable is used to update the velocity field and the pressure field. The pressure field is carefully updated by taking into account the velocity divergence field. This updating strategy can be rigorously proven to be able to eliminate the unphysical pressure boundary layer and is crucial for the correct temporal convergence rate. Our current staggered-mesh scheme is distinct from other conventional ones in that we store the velocity components at cell centers and the auxiliary variable at vertices. The fluid-interface is captured by solving an advection equation for the volume fraction of one of the fluids. The same matrixfree finite volume method as the one used for momentum equations is used to solve the advection equation. We will focus on the interface sharpening strategy to minimize the smearing of the interface over time. We have developed and implemented a global mass conservation algorithm which enforces the conservation of the mass for each fluid. ecently, we developed a hybrid finite element/volume (FE/FV) solver [1] for incompressible flows. The hybrid solver is based on the well-known pressure correction (projection) method [2, 3]. The solution procedure follows a segregated approach to decouple the pressure from the velocity. The velocity field is updated by solving the momentum equation provided that a known pressure field is given as a source term, through a cell-centered finite volume (FV) discretization. The pressure does not directly enter the momentum equation. Instead, an auxiliary variable, which is closely related to the pressure, takes the place of pressure in the momentum equation, providing pressure gradient information. We put the auxiliary variable on the vertices of cells. This deployment provides a convenient way to evaluate the pressure gradient using the local finite element basis functions. The incremental value of the auxiliary variable is computed by solving a Poisson equation using the Galerkin finite element (FE) method. The auxiliary variable is then used to update the velocity field. After the final velocity field is determined, the pressure can be updated using the auxiliary variable and the velocity divergence field. The pressure is updated in such a way that the pressure field is free of unphysical conditions in the boundary layer. Our hybrid finite volume/element solver is aimed to take advantage of the merits of both the FV and the FE methods and avoid their shortcomings. For example, highly-stretched cells (also known as high-aspect-ratio cells) are commonly used inside the boundary layer for high Reynolds number flows to resolve the boundary layer and reduce the number of cells. The stabilization parameters in the stabilized FE based flow solvers [4, 5] are related to the characteristic element length that is not well defined for high-aspect-ratio mesh elements. Due to this, it is very difficult to control the numerical dissipation of stabilized finite element solvers. By contrast, the finite volume flow